Techniques for synthesis of birefringent filters have been extensively described in the literature, as for example in "Optical Network Synthesis Using Birefringent Crystals. I. Synthesis of Lossless Networks of Equal-Length Crystals", 54 J. Opt. Soc. Am. 1267 (1964), by S. E. Harris, E. O. Amman, I. C. Chang; "Optical Network Synthesis Using Birefringent Crystals. III. Some General Properties of Lossless Birefringent Networks", 56 J. Opt. Soc. Am. 943 (1966), by E. O. Amman; "Optical Network Synthesis Using Birefringent Crystals. IV. Synthesis of Lossless Double-Pass Networks", 56 J. Opt. Soc. Am. (7), 952 (1966), by E. O. Amman; and "Synthesis of Optical Birefringent Networks", Progress in Optics IX 1971, pp. 123-177 (North-Holland, Amsterdam) by E. O. Amman. The birefringent filters disclosed in the literature consist of N optical retarders placed in series and positioned between an entrance polarizer and an exit polarizer, with each element oriented at a specific angle. Taking the transmission axis of the entrance polarizer as 0.degree., each retarder element i has its fast axis at an angle .phi..sub.i, and the exit polarizer is oriented at angle .phi..sub.p. The optical retardance may be different for each element, and together the retardances form a set R.sub.i with N members.
The prior art describes methods, based on time impulse-response analysis, for determining the values .phi..sub.i and R.sub.i so that the resultant filter has a desired response. The resulting filter response is an N-point harmonic function, where the harmonic coefficients are selected to achieve some particular optical response. Often, coefficients are chosen to yield a best approximation to some desired function, and are selected by, e.g., equating the harmonic coefficients to the terms in a Fourier expansion of the desired function, or by some similar method.
The synthesis procedure utilizes a set of harmonic coefficients C.sub.i and provides one or more sets of angles .phi..sub.i that will achieve the desired filter response. In general, the solutions are not unique, and there can be several equivalent sets of angles .phi..sub.i which yield the desired filter action. When this occurs, it is possible to construct a filter in one of several ways, depending on which solution set of angles .phi..sub.i is chosen. This procedure thus provides a powerful synthesis technique for developing optical filters with a wide variety of bandpass and filtration shapes.
Although in this analytical procedure each of the retarders is considered as having the same retardance R, this does not lead to a loss in generality. For example, one may construct a retarder of value 4R by placing four retarders of value R in series. A set of retarders with arbitrary rational retardance values may thus be built up from a number of equal-value retarders. The selection of equal-value retarders in the analytical procedure is essentially one of notation, rather than one of practical limitation.
Others have employed the birefringent filter synthesis method to design filters with equal passband and stopband widths for use in optical multiplexers; see, e.g., "Flat Passband Birefringent Wavelength Domain Multiplexer", 23 Electronics Letters 106-7 (1987), by W. J. Carlsen and C. F. Buhrer. This synthesis procedure permits independent choice of passband and stopband ripple by selection of the harmonic coefficients, which in turn determine the angles .phi..sub.i. Edge sharpness is a function of the number of retarder elements, with more retarder elements yielding greater sharpness.
U.S. Pat. No. 4,239,349 to Scheffer describes the use of birefringent films in conjunction with twisted-nematic cells and a neutral linear polarizer to achieve color switching. A birefringent filter may be used together with an optical polarization switch, such as a twisted-nematic liquid crystal cell, to create a switchable filter which can select between a certain filter state and its complement. Such a filter is shown by way of example in FIG. 1. The switch element is placed immediately adjacent to the entrance or exit polarizer, and effectively rotates it by 90.degree. or, in its alternate state, not at all. Hence the two filter states are a color and its complement. Similar results may be achieved with liquid crystal half-wave plates or ferroelectric liquid crystal cells. The switch member may also incorporate retarder elements to achieve its switching action, as is described in "A New RGB Tunable Filter Technology" 2650 Proc. of S.P.I.E., 98 (1996), by Sharp and Johnson. However, the essential function of such retarders is to selectively rotate the plane of polarization as part of the switch, rather than to define a passband as part of the birefringent filter.
On the other hand, one may also rotate individual retarder elements to change the orientation angles .phi..sub.i and thereby alter the filter response. However, such a system would be expensive and mechanically intricate.
Buhrer has described how mechanically rotating elements may be used to tune a multiplexer to all its possible settings ("Synthesis and Tuning of High-Order Solc-Type Birefringent Filters", 33 Applied Optics 2249-54 (1994)). Electro-optical methods have been explored in U.S. Pat. No. 4,157,008 to Pinnow et al. using a stack of doped crystals. A similar approach is possible using a series of planar-aligned smectic A* liquid crystal cells, which exhibit fixed retardance with a voltage-adjustable crystal orientation. Neither approach, however, has been widely used to date because of the limited crystal orientation ranges available.
Amman described a method for synthesizing filters that may be used in a double-pass mode. These filters include a mirror disposed at one end of a series of N retarder elements so that light passes through the retarders twice: once on its way in toward the mirror, and the second time on its way out after reflection. This arrangement is optically equivalent to a stack of 2N retarders with their fast axes oriented at angles {.phi..sub.1, .phi..sub.2, . . . .phi..sub.N-1, .phi..sub.N, .phi..sub.N, .phi..sub.N-1, . . . .phi..sub.2, .phi..sub.1 }.
In some designs, polarization couplers are used so that light is admitted into, and extracted from, the retarder stack without interference between the two beams. Although not all filters can be constructed in this way since the harmonic coefficients of the filter response must meet certain criteria to realize this design, it is possible in many cases to employ this double-pass scheme to achieve the resolution of a filter with 2N retarders while actually using only N such components.
Amman has written various theorems pertaining to networks of retarders, along with proofs. These are useful in understanding and designing such filters, and will be referred to below.
The use of a pair of high-order retarder elements between crossed polarizers at 0.degree. and 90.degree. has been described by Evans in "The Birefringent Filter", 39 J. Opt. Soc. Am. 229-42 (1949). These elements have their axes crossed, at .+-.45.degree. and -45.degree.. A retarder with its axis at 0.degree. (or 90.degree.) is placed between the elements; this middle retarder can be a simple .lambda./2 plate, or it may be a retarder which contributes to defining a passband constructed as an (N+1/2).lambda. order waveplate. The positioning of the intermediate retarder between the outer two members dispenses with the need for a separate polarizer associated with the intermediate element thus reducing the number of polarizers and increasing the filter transmission relative to the usual Lyot design. The former arrangement is termed a wide-field filter, and the latter a split-element filter. Evans also considered adding further retarders about the split-element stage to create a doubly-split stage. In all cases, the retarders have their fast axes oriented at an angle of 0.degree., +/-45.degree., or 90.degree. relative to the entrance polarizer.
Tunable versions of the wide-field filters are taught in U.S. Pat. No. 5,247,378 to Miller, with continuously variable nematic liquid crystal retarders as the tuning element.
U.S. Pat. No. 5,528,393 to Sharp and Johnson describes both wide-field and split-element filters which use ferroelectric smectic C* or smectic A* switches to obtain a discretely or continuously tunable bandpass, respectively. These switches have a fixed retardance and an electrically switchable crystal orientation. Sharp and Johnson also describe a reflection-mode split element filter where a mirror is placed after the central element so that light traverses the elements in both directions. This system is said to allow the same function to be achieved using a reduced number of components.
At present, however, no birefringent filter allows for selection between a white-state in which all wavelengths are transmitted and a desired complex filtration state. Nor is there a practical birefringent filter which offers non-mechanical switching between several complex transmission states, other than between a filter response and its complement.